Notes 7: Knowledge Representation, The Propositional Calculus ICS 171 Fall 2006

Outline Knowledge-based agents Wumpus world Logic in general - models and entailment Propositional (Boolean) logic Equivalence, validity, satisfiability Inference rules and theorem proving forward chaining backward chaining resolution

Knowledge bases Knowledge base = set of sentences in a formal language Declarative approach to building an agent (or other system): Tell it what it needs to know Then it can Ask itself what to do - answers should follow from the KB Agents can be viewed at the knowledge level i.e., what they know, regardless of how implemented Or at the implementation level i.e., data structures in KB and algorithms that manipulate them

A simple knowledge-based agent The agent must be able to: Represent states, actions, etc. Incorporate new percepts Update internal representations of the world Deduce hidden properties of the world Deduce appropriate actions

Wumpus World PEAS description Performance measure gold +1000, death -1000 -1 per step, -10 for using the arrow Environment Squares adjacent to wumpus are smelly Squares adjacent to pit are breezy Glitter iff gold is in the same square Shooting kills wumpus if you are facing it Shooting uses up the only arrow Grabbing picks up gold if in same square Releasing drops the gold in same square Sensors: Stench, Breeze, Glitter, Bump, Scream Actuators: Left turn, Right turn, Forward, Grab, Release, Shoot

Wumpus world characterization Fully Observable No – only local perception Deterministic Yes – outcomes exactly specified Episodic No – sequential at the level of actions Static Yes – Wumpus and Pits do not move Discrete Yes Single-agent? Yes – Wumpus is essentially a natural feature

Exploring a wumpus world

Exploring a wumpus world

Exploring a wumpus world

Exploring a wumpus world

Exploring a wumpus world

Exploring a wumpus world

Exploring a wumpus world

Exploring a wumpus world

Logic in general Logics are formal languages for representing information such that conclusions can be drawn Syntax defines the sentences in the language Semantics define the "meaning" of sentences; i.e., define truth of a sentence in a world E.g., the language of arithmetic x+2 ≥ y is a sentence; x2+y > {} is not a sentence x+2 ≥ y is true iff the number x+2 is no less than the number y x+2 ≥ y is true in a world where x = 7, y = 1 x+2 ≥ y is false in a world where x = 0, y = 6

Entailment Entailment means that one thing follows from another: KB ╞ α Knowledge base KB entails sentence α if and only if α is true in all worlds where KB is true E.g., the KB containing “the Giants won” and “the Reds won” entails “Either the Giants won or the Reds won” E.g., x+y = 4 entails 4 = x+y Entailment is a relationship between sentences (i.e., syntax) that is based on semantics

Models Logicians typically think in terms of models, which are formally structured worlds with respect to which truth can be evaluated We say m is a model of a sentence α if α is true in m M(α) is the set of all models of α Then KB ╞ α iff M(KB)  M(α) E.g. KB = Giants won and Reds won α = Giants won

Entailment in the wumpus world Situation after detecting nothing in [1,1], moving right, breeze in [2,1] Consider possible models for KB assuming only pits 3 Boolean choices  8 possible models

Wumpus models

Wumpus models KB = wumpus-world rules + observations

Wumpus models KB = wumpus-world rules + observations α1 = "[1,2] is safe", KB ╞ α1, proved by model checking

Wumpus models KB = wumpus-world rules + observations

Wumpus models KB = wumpus-world rules + observations α2 = "[2,2] is safe", KB ╞ α2

Inference KB ├i α = sentence α can be derived from KB by procedure i Soundness: i is sound if whenever KB ├i α, it is also true that KB╞ α Completeness: i is complete if whenever KB╞ α, it is also true that KB ├i α Preview: we will define a logic (first-order logic) which is expressive enough to say almost anything of interest, and for which there exists a sound and complete inference procedure. That is, the procedure will answer any question whose answer follows from what is known by the KB.

Propositional logic: Syntax Propositional logic is the simplest logic – illustrates basic ideas The proposition symbols P1, P2 etc are sentences If S is a sentence, S is a sentence (negation) If S1 and S2 are sentences, S1  S2 is a sentence (conjunction) If S1 and S2 are sentences, S1  S2 is a sentence (disjunction) If S1 and S2 are sentences, S1  S2 is a sentence (implication) If S1 and S2 are sentences, S1  S2 is a sentence (biconditional)

Propositional logic: Semantics Each model specifies true/false for each proposition symbol E.g. P1,2 P2,2 P3,1 false true false With these symbols, 8 possible models, can be enumerated automatically. Rules for evaluating truth with respect to a model m: S is true iff S is false S1  S2 is true iff S1 is true and S2 is true S1  S2 is true iff S1is true or S2 is true S1  S2 is true iff S1 is false or S2 is true i.e., is false iff S1 is true and S2 is false S1  S2 is true iff S1S2 is true andS2S1 is true Simple recursive process evaluates an arbitrary sentence, e.g., P1,2  (P2,2  P3,1) = true  (true  false) = true  true = true

Truth tables for connectives

Wumpus world sentences Let Pi,j be true if there is a pit in [i, j]. Let Bi,j be true if there is a breeze in [i, j].  P1,1 B1,1 B2,1 "Pits cause breezes in adjacent squares" B1,1  (P1,2  P2,1) B2,1  (P1,1  P2,2  P3,1)

Truth tables for inference

Inference by enumeration Depth-first enumeration of all models is sound and complete For n symbols, time complexity is O(2n), space complexity is O(n)

Logical equivalence Two sentences are logically equivalent} iff true in same models: α ≡ ß iff α╞ β and β╞ α

Validity and satisfiability A sentence is valid if it is true in all models, e.g., True, A A, A  A, (A  (A  B))  B Validity is connected to inference via the Deduction Theorem: KB ╞ α if and only if (KB  α) is valid A sentence is satisfiable if it is true in some model e.g., A B, C A sentence is unsatisfiable if it is true in no models e.g., AA Satisfiability is connected to inference via the following: KB ╞ α if and only if (KB α) is unsatisfiable

Proof methods Proof methods divide into (roughly) two kinds: Application of inference rules Legitimate (sound) generation of new sentences from old Proof = a sequence of inference rule applications Can use inference rules as operators in a standard search algorithm Typically require transformation of sentences into a normal form Model checking truth table enumeration (always exponential in n) improved backtracking, e.g., Davis--Putnam-Logemann-Loveland (DPLL) heuristic search in model space (sound but incomplete) e.g., min-conflicts-like hill-climbing algorithms

li  …  li-1  li+1  …  lk  m1  …  mj-1  mj+1 ...  mn Resolution Conjunctive Normal Form (CNF) conjunction of disjunctions of literals clauses E.g., (A  B)  (B  C  D) Resolution inference rule (for CNF): li …  lk, m1  …  mn li  …  li-1  li+1  …  lk  m1  …  mj-1  mj+1 ...  mn where li and mj are complementary literals. E.g., P1,3  P2,2, P2,2 P1,3 Resolution is sound and complete for propositional logic

Resolution Soundness of resolution inference rule: (li  …  li-1  li+1  …  lk)  li mj  (m1  …  mj-1  mj+1 ...  mn) (li  …  li-1  li+1  …  lk)  (m1  …  mj-1  mj+1 ...  mn)

Conversion to CNF B1,1  (P1,2  P2,1) Eliminate , replacing α  β with (α  β)(β  α). (B1,1  (P1,2  P2,1))  ((P1,2  P2,1)  B1,1) 2. Eliminate , replacing α  β with α β. (B1,1  P1,2  P2,1)  ((P1,2  P2,1)  B1,1) 3. Move  inwards using de Morgan's rules and double-negation: (B1,1  P1,2  P2,1)  ((P1,2  P2,1)  B1,1) 4. Apply distributivity law ( over ) and flatten: (B1,1  P1,2  P2,1)  (P1,2  B1,1)  (P2,1  B1,1)

Resolution algorithm Proof by contradiction, i.e., show KBα unsatisfiable

Resolution example KB = (B1,1  (P1,2 P2,1))  B1,1 α = P1,2

Rules of inference

Resolution in Propositional Calculus Using clauses as wffs Literal, clauses, conjunction of clauses (cnfs) Resolution rule: Resolving (P V Q) and (P V  Q) P Generalize modus ponens, chaining . Resolving a literal with its negation yields empty clause. Resolution is sound Resolution is NOT complete: P and R entails P V R but you cannot infer P V R From (P and R) by resolution Resolution is complete for refutation: adding (P) and (R) to (P and R) we can infer the empty clause. Decidability of propositional calculus by resolution refutation: if a wff w is not entailed by KB then resolution refutation will terminate without generating the empty clause.

Converting wffs to Conjunctive clauses 1. Eliminate implications 2. Reduce the scope of negation sign 3. Convert to cnfs using the associative and distributive laws

Soundness of resolution

The party example If Alex goes, then Beki goes: A  B If Chris goes, then Alex goes: C  A Beki does not go: not B Chris goes: C Query: Is it possible to satisfy all these conditions? Should I go to the party?

Example of proof by Refutation Assume the claim is false and prove inconsistency: Example: can we prove that Chris will not come to the party? Prove by generating the desired goal. Prove by refutation: add the negation of the goal and prove no model Proof: Refutation:

The moving robot example bat_ok,liftable moves ~moves, bat_ok

Proof by refutation Given a database in clausal normal form KB Find a sequence of resolution steps from KB to the empty clauses Use the search space paradigm: States: current cnf KB + new clauses Operators: resolution Initial state: KB + negated goal Goal State: a database containing the empty clause Search using any search method

Proof by refutation (contd.) Or: Prove that KB has no model - PSAT A cnf theory is a constraint satisfaction problem: variables: the propositions domains: true, false constraints: clauses (or their truth tables) Find a solution to the csp. If no solution no model. This is the satisfiability question Methods: Backtracking arc-consistency  unit resolution, local search

Complexity of propositional inference Checking truth tables is exponential Satisfiability is NP-complete However, frequently generating proofs is easy. Propositional logic is monotonic If you can entail alpha from knowledge base KB and if you add sentences to KB, you can infer alpha from the extended knowledge-base as well. Inference is local Tractable Classes: Horn, 2-SAT Horn theories: Q <-- P1,P2,...Pn Pi is an atom in the language, Q can be false. Solved by modus ponens or “unit resolution”.

Forward and backward chaining Horn Form (restricted) KB = conjunction of Horn clauses Horn clause = proposition symbol; or (conjunction of symbols)  symbol E.g., C  (B  A)  (C  D  B) Modus Ponens (for Horn Form): complete for Horn KBs α1, … ,αn, α1  …  αn  β β Can be used with forward chaining or backward chaining. These algorithms are very natural and run in linear time

Forward chaining Idea: fire any rule whose premises are satisfied in the KB, add its conclusion to the KB, until query is found

Forward chaining algorithm Forward chaining is sound and complete for Horn KB

Forward chaining example

Forward chaining example

Forward chaining example

Forward chaining example

Forward chaining example

Forward chaining example

Forward chaining example

Forward chaining example

Proof of completeness FC derives every atomic sentence that is entailed by KB FC reaches a fixed point where no new atomic sentences are derived Consider the final state as a model m, assigning true/false to symbols Every clause in the original KB is true in m a1  …  ak  b Hence m is a model of KB If KB╞ q, q is true in every model of KB, including m

Backward chaining Idea: work backwards from the query q: to prove q by BC, check if q is known already, or prove by BC all premises of some rule concluding q Avoid loops: check if new subgoal is already on the goal stack Avoid repeated work: check if new subgoal has already been proved true, or has already failed

Backward chaining example

Backward chaining example

Backward chaining example

Backward chaining example

Backward chaining example

Backward chaining example

Backward chaining example

Backward chaining example

Backward chaining example

Backward chaining example

Forward vs. backward chaining FC is data-driven, automatic, unconscious processing, e.g., object recognition, routine decisions May do lots of work that is irrelevant to the goal BC is goal-driven, appropriate for problem-solving, e.g., Where are my keys? How do I get into a PhD program? Complexity of BC can be much less than linear in size of KB

Efficient propositional inference Two families of efficient algorithms for propositional inference: Complete backtracking search algorithms DPLL algorithm (Davis, Putnam, Logemann, Loveland) Incomplete local search algorithms WalkSAT algorithm

The DPLL algorithm Determine if an input propositional logic sentence (in CNF) is satisfiable. Improvements over truth table enumeration: Early termination A clause is true if any literal is true. A sentence is false if any clause is false. Pure symbol heuristic Pure symbol: always appears with the same "sign" in all clauses. e.g., In the three clauses (A  B), (B  C), (C  A), A and B are pure, C is impure. Make a pure symbol literal true. Unit clause heuristic Unit clause: only one literal in the clause The only literal in a unit clause must be true.

The DPLL algorithm

The WalkSAT algorithm Incomplete, local search algorithm Evaluation function: The min-conflict heuristic of minimizing the number of unsatisfied clauses Balance between greediness and randomness

The WalkSAT algorithm

Hard satisfiability problems Consider random 3-CNF sentences. e.g., (D  B  C)  (B  A  C)  (C  B  E)  (E  D  B)  (B  E  C) m = number of clauses n = number of symbols Hard problems seem to cluster near m/n = 4.3 (critical point)

Hard satisfiability problems

Hard satisfiability problems Median runtime for 100 satisfiable random 3-CNF sentences, n = 50

Inference-based agents in the wumpus world A wumpus-world agent using propositional logic: P1,1 W1,1 Bx,y  (Px,y+1  Px,y-1  Px+1,y  Px-1,y) Sx,y  (Wx,y+1  Wx,y-1  Wx+1,y  Wx-1,y) W1,1  W1,2  …  W4,4 W1,1  W1,2 W1,1  W1,3 …  64 distinct proposition symbols, 155 sentences

Summary Logical agents apply inference to a knowledge base to derive new information and make decisions Basic concepts of logic: syntax: formal structure of sentences semantics: truth of sentences wrt models entailment: necessary truth of one sentence given another inference: deriving sentences from other sentences soundness: derivations produce only entailed sentences completeness: derivations can produce all entailed sentences Wumpus world requires the ability to represent partial and negated information, reason by cases, etc. Resolution is complete for propositional logic Forward, backward chaining are linear-time, complete for Horn clauses Propositional logic lacks expressive power